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Theorem mhmrcl2 15590
Description: Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
Assertion
Ref Expression
mhmrcl2  |-  ( F  e.  ( S MndHom  T
)  ->  T  e.  Mnd )

Proof of Theorem mhmrcl2
Dummy variables  f 
s  t  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mhm 15586 . 2  |- MndHom  =  ( s  e.  Mnd , 
t  e.  Mnd  |->  { f  e.  ( (
Base `  t )  ^m  ( Base `  s
) )  |  ( A. x  e.  (
Base `  s ) A. y  e.  ( Base `  s ) ( f `  ( x ( +g  `  s
) y ) )  =  ( ( f `
 x ) ( +g  `  t ) ( f `  y
) )  /\  (
f `  ( 0g `  s ) )  =  ( 0g `  t
) ) } )
21elmpt2cl2 6419 1  |-  ( F  e.  ( S MndHom  T
)  ->  T  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803   ` cfv 5529  (class class class)co 6203    ^m cmap 7327   Basecbs 14295   +g cplusg 14360   0gc0g 14500   Mndcmnd 15531   MndHom cmhm 15584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-dm 4961  df-iota 5492  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-mhm 15586
This theorem is referenced by:  mhmf1o  15595  resmhm  15609  mhmco  15612  mhmima  15613  pwsco2mhm  15621  gsumwmhm  15645  mhmmulg  15781  mhmhmeotmd  26522
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